Associate space with respect to a semi-finite measure : master's thesis
Talimdjioski, Filip (Author), Kandić, Marko (Mentor) More about this mentor... This link opens in a new window

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We introduce the basic theory of vector lattices and order bounded operators. Order continuous operators, the order dual space, normed lattices and several variants of the Fatou property of normed lattices are studied. We then introduce normed function spaces, saturated function seminorms and the associate space to a normed function space, as well as semi-finite and localizable measures. We prove that the associate space $E'$ of an arbitrary saturated Banach function space $E$ with respect to a semi-finite measure $\mu$ equals the order continuous dual space $E_n^{^\sim}$ if and only if $E'$ has the strong Fatou property. If $E$ is furthermore $\sigma$-order continuous we prove that $E_n^{^\sim}$, the sigma-order continuous dual space $E_c^{^\sim}$ and the norm dual $E^*$ are equal which implies that in the previously mentioned result the equality $E' = E_n^{^\sim}$ can be replaced with $E' = E^*$. Also, if $\mu$ is localizable, we prove that $E'$ has the strong Fatou property which implies that $E' = E_n^{^\sim}$. Finally, we give an example where the aforementioned equality fails.

Keywords:semi-finite and localizable measures, Banach function spaces, associate space, order and $\sigma$-order continuity, Fatou property
Work type:Master's thesis/paper (mb22)
Organization:FMF - Faculty of Mathematics and Physics
COBISS.SI-ID:18718297 This link opens in a new window
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Secondary language

Title:Pridruženi prostor glede na semi-končno mero
Uvedemo osnovno teorijo vektorskih mrež in urejenostno omejenih operatorjev. Preučujemo urejenostno zvezne operatorje, urejenostni dualni prostor normiranih mrež in več različic Fatoujeve lastnosti normiranih mrež. Nato uvedemo normirane funkcijske prostore, nasičene funkcijske polnorme in pridruženi prostor normiranemu funkcijskemu prostoru, na koncu pa tudi semi-končne in lokalizabilne mere. Dokažemo, da je pridruženi prostor $E'$ poljubnega nasičenega Banachovega funkcijskega prostora $E$ glede na semi-končno mero $\mu$ enak urejenostno zveznemu dualnemu prostoru $E_n^{^\sim}$ natanko tedaj, ko ima $E'$ krepko Fatoujevo lastnost. Če je $E$ nadalje sigma-urejenostno zvezen, dokažemo, da so $E_n^{^\sim}$, $\sigma$-urejenostno zvezni dualni prostor $E_c^{^\sim}$ in normirani dualni prostor $E^*$ enaki, kar pomeni, da lahko v prej omenjenem rezultatu enakost $E' = E_n^{^\sim}$ nadomestimo z $E' = E^*$. Če je $\mu$ lokalizabilna, dokažemo, da ima $E'$ krepko Fatoujevo lastnost, kar pomeni, da je $E' = E_n^{^\sim}$. Na koncu navedemo primer, ko pravkar omenjena enakost ne drži.

Keywords:semi-končne in lokalizabilne mere, Banachovi funkcijski prostori, pridruženi prostor, urejenostna in $\sigma$-urejenostna zveznost, Fatoujeva lastnost

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